Combining numerical index laws | Year 8

01 / LEARN

The idea, step by step

Write a plan

First identify the common base and the operation signs. Simplify powers of powers, then combine products and quotients. If every factor uses one base, a single exponent calculation can organise the work. Ordinary addition must stay separate: $2^{3} + 2^{4}$ is not a product, so its exponents cannot simply be added.

Check without repeating the shortcut

For a small expression, evaluate the original powers separately and compare with your simplified answer. This checks the result using a different route. Leave a large result as one power when asked, and evaluate only when requested. These lessons use numerical bases and nonnegative final powers; later study extends the ideas to negative indices and variables.

A common trap

Every operation on powers uses the same exponent shortcut.

Instead: Products add exponents; quotients subtract; powers of powers multiply. A sum must be evaluated term by term.

02 / WORKED EXAMPLES

See the thinking

Example 1

Power, product and quotient

Simplify $\frac{{(2^{3})}^{2} \times 2^{4}}{2^{5}}$ and evaluate.

${(2^{3})}^{2} = 2^{6}$ .
The final exponent is $6 + 4 - 5 = 5$ .
$2^{5} = 32$ .

2^{5} = 32

Example 2

A different base

Simplify $\frac{3^{2} \times {(3^{3})}^{2}}{3^{5}}$ and evaluate.

${(3^{3})}^{2} = 3^{6}$ .
The exponent is $2 + 6 - 5 = 3$ .
$3^{3} = 27$ .

3^{3} = 27

Check your understanding

Simplify $4^{3} \div 4^{2}$ .

Show answer

$4^{3 - 2} = 4^{1} = 4$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $\frac{{(5^{2})}^{3}}{5^{4}}$ and evaluate.
2 marks
Simplify $\frac{2^{3} \times 2^{4}}{2^{5}}$ and evaluate.
2 marks
Simplify $\frac{{(10^{2})}^{3} \times 10^{0}}{10^{4}}$ and evaluate.
2 marks
Simplify $\frac{{(4^{3} \div 4)}^{2}}{4^{2}}$ and evaluate.
3 marks
Evaluate ${(6^{2})}^{2} \div 6^{4}$ .
2 marks
Explain the error in ${(2^{3})}^{2} = 2^{5}$ and give the correct value.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Simplify $\frac{{(5^{2})}^{3}}{5^{4}}$ and evaluate.
2 marks
The exponent is $2 \times 3 - 4 = 2$ .
$5^{2} = 25$
Simplify $\frac{2^{3} \times 2^{4}}{2^{5}}$ and evaluate.
2 marks
$3 + 4 - 5 = 2$ .
$2^{2} = 4$
Simplify $\frac{{(10^{2})}^{3} \times 10^{0}}{10^{4}}$ and evaluate.
2 marks
$2 \times 3 + 0 - 4 = 2$ .
$10^{2} = 100$
Simplify $\frac{{(4^{3} \div 4)}^{2}}{4^{2}}$ and evaluate.
3 marks
$4^{3} \div 4 = 4^{2}$ .
${(4^{2})}^{2} \div 4^{2} = 4^{4 - 2} = 4^{2} = 16$ .
$4^{2} = 16$
Evaluate ${(6^{2})}^{2} \div 6^{4}$ .
2 marks
$6^{4 - 4} = 6^{0} = 1$ .
1
Explain the error in ${(2^{3})}^{2} = 2^{5}$ and give the correct value.
3 marks
${(2^{3})}^{2} = 2^{3 \times 2} = 2^{6} = 64$ .
The exponents multiply; the correct value is 64.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Exponent notation review; multiplying same-base powers; dividing same-base powers; powers of powers; zero index; combining numerical index laws. Negative indices and general variable index laws become Year 9 core.

Official curriculum source ↗

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